Related papers: A map of contour integral-based eigensolvers for s…
Novel contour descriptors, called eigencontours, based on low-rank approximation are proposed in this paper. First, we construct a contour matrix containing all object boundaries in a training set. Second, we decompose the contour matrix…
In this work, the infinite GMRES algorithm, recently proposed by Correnty et al., is employed in contour integral-based nonlinear eigensolvers, avoiding the computation of costly factorizations at each quadrature node to solve the linear…
The eigenvalue density of a matrix plays an important role in various types of scientific computing such as electronic-structure calculations. In this paper, we propose a quantum algorithm for computing the eigenvalue density in a given…
Recently, three numerical methods for the computation of eigenvalues of singular matrix pencils, based on a rank-completing perturbation, a rank-projection, or an augmentation were developed. We show that all three approaches can be…
Solving large-scale Generalized Eigenvalue Problems (GEPs) is a fundamental yet computationally prohibitive task in science and engineering. As a promising direction, contour integral (CI) methods, such as the CIRR algorithm, offer an…
Various iterative eigenvalue solvers have been developed to compute parts of the spectrum for a large sparse matrix, including the power method, Krylov subspace methods, contour integral methods, and preconditioned solvers such as the so…
We consider the numerical computation of resonances for metallic grating structures with dispersive media and small slit holes. The underlying eigenvalue problem is nonlinear and the mathematical model is multiscale due to the existence of…
Generalized eigenvalue problems involving a singular pencil may be very challenging to solve, both with respect to accuracy and efficiency. While Part I presented a rank-completing addition to a singular pencil, we now develop two…
Investigating the stability of nonlinear waves often leads to linear or nonlinear eigenvalue problems for differential operators on unbounded domains. In this paper we propose to detect and approximate the point spectra of such operators…
In recent years, contour-based eigensolvers have emerged as a standard approach for the solution of large and sparse eigenvalue problems. Building upon recent performance improvements through non-linear least square optimization of…
This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems…
Often the easiest way to discretize an ordinary or partial differential equation is by a rectangular numerical method, in which n basis functions are sampled at m>>n collocation points. We show how eigenvalue problems can be solved in this…
Leading eigenvalue problems for large scale matrices arise in many applications. Coordinate-wise descent methods are considered in this work for such problems based on a reformulation of the leading eigenvalue problem as a non-convex…
Contour integration schemes are a valuable tool for the solution of difficult interior eigenvalue problems. However, the solution of many large linear systems with multiple right hand sides may prove a prohibitive computational expense. The…
Convex optimization encompasses a wide range of optimization problems that contain many efficiently solvable subclasses. Interior point methods are currently the state-of-the-art approach for solving such problems, particularly effective…
This paper proposes a rational filtering domain decomposition technique for the solution of large and sparse symmetric generalized eigenvalue problems. The proposed technique is purely algebraic and decomposes the eigenvalue problem…
Eigenvalue problems are fundamental to mathematics and science. We present a simple algorithm for determining eigenvalues and eigenfunctions of the Laplace--Beltrami operator on rather general curved surfaces. Our algorithm, which is based…
This paper presents a method for computing eigenvalues and eigenvectors for some types of nonlinear eigenvalue problems. The main idea is to approximate the functions involved in the eigenvalue problem by rational functions and then apply a…
In this article, we study eigenvalue problems associated to self-adjoint operators and their approximation obtained by subspace projection, as used in the reduced basis method for instance. We provide error bounds between the exact…
A fully numerical method to calculate loop integrals, a numerical contour-integration method, is proposed. Loop integrals can be interpreted as a contour integral in a complex plane for an integrand with multi-poles in the plane. Stable and…